# A calculus of the absurd

##### 17.2.2 The properties which make modular arithmetic

The first core property is this

• Theorem 17.2.2 If $$a \equiv _m b$$ and $$c \equiv _m d$$, then

$$a + c \equiv _m b + d$$

Proof: TODO

• Theorem 17.2.3 If $$a \equiv _m b$$ and $$c \equiv _m d$$, then

$$ac \equiv _m cd$$

Proof: TODO

• Example 17.2.1 Calculate

$$R_{10}(17^{17})$$

This requires a bit of careful thinking about, but the result should be

\begin{align} 17^{17} &\equiv _{10} 7^{17} &\equiv _{10} 7 7^{16} \\ &\equiv _{10} 7 \left (7^2\right )^8 \\ &\equiv _{10} 7 \left (-1\right )^{8} \\ &\equiv _{10} 7 \end{align}